A spline is a piecewise polynomial function on \([a, b] \subset \mathbb{R}\). The points \(t_0, t_1, \dots, t_n\) at which the polynomial segments are joined are called knots.

\[ S(x) = \begin{cases} P_0(x): t \in [t_0, t_1) \\ P_1(x): t \in [t_1, t_2) \\ \cdots \\ P_{n-1}(x): t \in [t_{n-1}, t_n] \end{cases} \]

If \(P_i\) are polynomials of degree ≤ m, then the spline S has degree m and order m+1.

A spline is continuous in all derivatives everywhere except at its knots. At its knots a spline can have continuity order ranging from -1 (no continuity) through m-1.

Splines most common in practical use are those with degree m and continuity order m-1.

For a given knot vector \(\vec{t} = (t_0, \dots t_n)\), and a given spline degree m, there exists a unique set of splines (basis splines) meeting the following criteria:

• Locality: Basis spline \(B_i\) is zero outside of the segment \([t_i, t_{i+n}]\)
• Scaling: \(\sum B_i(x) = 1, \forall x \in [t_m, t_{n-m}]\)

All splines of degree m and continuity order m-1 over the knot vector \(\vec t\) can be expressed as a linear combination of basis splines.

Think of basis splines as blending functions that provide smooth transitions between fixed control points.

Control points shape the spline.

Control points shape the spline. In a spline model, model parameters are (related to) control points.

In simple a simple B-spline model, quality of fit is measured by mean square error (MSE).

B-spline fitting with MSE is sensitive to number of knots. Too many knots lead to overfitting.

One solution to this problem is to include smoothness in measure of fit; this is known as “penalty”.

Most common penalty is integral of second derivative squared, also knows as 2nd order penalty.

B-spline used for fitting with a penalty term are known as penalized B-splines, or P-splines.

B-splines make no smoothness guarantee at the beginning and end of a cyclic time period.

That is what cyclic B-splines are used for